Fuzzy Logic & Probability Theory: 
模糊邏輯與概率論
                        
Clarification towards Building a Bridge
                        

                        Fuzzy logic and probability theory are the most powerful tools to overcome 
the imperfection (see Fig.1). Fuzzy logic is mainly responsible for 
representation and processing of vague data (ill-defined, fuzzy). Probability 
theory is mainly responsible for representation and processing of uncertainty 
(randomness).
                        

                        
  
                        
  Fig.1. Imperfection and theories to handle it.

                        

                        Following table clarifies the differences between the two theories.
                        

                        
  
    
                        
      
        
                        Probability Measure
                        
                              		
      
        
                        Membership Function
                        
                              		
    
                        
    
                        
      
        
                        Calculates the probability that an
                        
        ill-known variable x ranging on U hits the well-known set A
                        
                              		
      
        
                        Calculates the membership of a
                        
        well-known variable x ranging on U hits the ill-known set A
                        
        
                        
                              		
    
                        
    
                        
      
        
                        Before an event happens
                        
                              		
      
        
                        After it happened
                        
                              		
    
                        
    
                        
      
        
                        Measure Theory
                        
                              		
      
        
                        Set Theory
                        
                              		
    
                        
    
                        
      
        
                        Domain is 2U (Boolean Algebra)
                        
        
                        
                              		
      
        
                        Domain is [0,1]U (Cannot be a Boolean Algebra)
                        
                              		
    
                        
                          

                        

                        A Bridge
                        

                        

                        Consider the following statements:
                        

                        


                        In such cases (which are very usual in pattern recognition, for instance), we 
are interested in probability of an event that cannot be defined exactly. 
Therefore, the only sophisticated way is to calculate the probability of a fuzzy 
event represented by a fuzzy set:
                        

                        
  
    
                        
      
      
        
                        Probability space
                        
                              		
    
                        
    
                        
      
      
        
                        Membership function
                        
                              		
      
    
                        
    
                        
      
      
        
                        The probability of the fuzzy event F
                        
                              		
    
                        
                          

                        

                        For more details see following papers/books:
                        

                          
  
                        1. 
    
                          Probability measures of fuzzy events, L.A.Zadeh, Journal Math. 
    Anal. Appl., vol 23, pp. 421-427, 1968
                          
  
                        2. 
    
                          Fuzzy Sets as a basis for a theory of possibility, L.A. Zadeh, 
    Fuzzy Sets and Systems, vol. 1, pp. 3-28, 1978
                          
  
                        3. 
    
                          Possibility Theory, D.Dubois, H. Prade, Plenum Press, 1988
                          
  
                        4. 
    
                          Fuzzy sets and probability : Misunderstandings, bridges and gaps, 
    D.Dubois, H. Prade, Proc. of the Second IEEE Inter. Conf. on Fuzzy Systems, 
    volume 2, pp. 1059-1068, 1993
                          
  
                          5. 

                        



                        

                              

                                  

                                    

                                              

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